Store - a store that sells many different items in different departments. Harrods is probably the world's best known stores.
Supermarket - a store that sells mainly food and household products.
Grocer (UK) / supermarket (US) - a store that sells food.
Greengrocer - sells fresh fruit and vegetables.
Butcher - sells fresh meat.
Baker - sells fresh bread and pastries.
Fish - fresh fish sold.
Chemical (UK) / Pharmacy (US) - sells medicines and toiletries.
Pharmacy (US) - selling drugs.
Kiosk - sells newspapers and magazines.
Stationer - sells paper products.
Optical - sells glasses / contact lenses.
DIY store - sells things for home improvement.
hardware shop / hardware / ironmonger - non-perishable, such as nails and screws products.
corner store (UK) - a store in the corner of your street, selling a range of staples - food, newspapers, candy, bread, etc.
delicatessen (deli) - selling specialist food usually is not found in supermarkets. For example, an Italian deli, an Asian deli.
library / bookstore - books.
market traders (people who work in a market) with stalls selling fruit and vegetables, clothing, household items and so on - market.
pet shop - for pets and pet food.
flea market - a group of stalls selling furniture or old clothes.
tea shop (United Kingdom) - like a cafe, but sells tea and cakes.
petrol station (United Kingdom) / station (US) sells gasoline, automobile products and sometimes food.